Question

For the traverse given below, the benchmark is at ground point L. The Reduced Level (RL) of L is 150.000 m. The direction of the traverse is from L to N. The readings on levelling staff using the same levelling instrument kept at Stations S1 and S2 during the traversing are shown in the figure.

Column 1	Column 2
	(P) 3.300 m
(I) Sum of Back Sights	(Q) 2.500 m
(II) Sum of Fore Sights	(R) 150.300 m
(III) RL of Point N	(S) 3.000 m
	(T) 150.500 m

Matching the information given under Column 1 and Column 2, which one of the following options is CORRECT?

• A. (I) - (P) ; (II) - (S) ; (III) - (R)
• B. (I) - (P) ; (II) - (Q) ; (III) - (R)
• C. (I) - (S) ; (II) - (P) ; (III) - (T)
• D. (I) - (S) ; (II) - (Q) ; (III) - (T)

Hint:

Use HI = RL + BS at each setup, then RL(next point) = HI - FS; M is the change point linking setup S1 to setup S2.

Answer 1

The Correct Option is A

Step 1: Understanding the Question.
This is a two-setup levelling (differential levelling) problem. The instrument is set up first at S1, where it reads L (a backsight, since L is the known benchmark being read from behind the instrument as it moves toward N) and M (a foresight, since the instrument is about to move past this point). Then the instrument moves to S2, where it reads M again (this time as a backsight, since M is now behind the direction of travel) and N (a foresight). M is called the change point, since it is sighted twice, once from each instrument position, and links the two setups together.

Step 2: Key Formula or Approach.
The two formulas needed are:
Height of Instrument, \(HI = RL_{known} + BS\) (add the backsight reading to the known point's level to get the height of the sight line).
Reduced Level of the next point, \(RL_{next} = HI - FS\) (subtract the foresight reading from the height of the sight line).
We are given \(RL_L = 150.000\) m, backsight readings \(BS_L=1.5\) m and \(BS_M=1.8\) m, and foresight readings \(FS_M=2.0\) m and \(FS_N=1.0\) m.

Step 3: Detailed Explanation.
First setup, at S1:
\[ HI_{S1} = RL_L + BS_L = 150.000 + 1.5 = 151.500\text{ m} \]
\[ RL_M = HI_{S1} - FS_M = 151.500 - 2.0 = 149.500\text{ m} \]
Second setup, at S2:
\[ HI_{S2} = RL_M + BS_M = 149.500 + 1.8 = 151.300\text{ m} \]
\[ RL_N = HI_{S2} - FS_N = 151.300 - 1.0 = 150.300\text{ m} \]
Sum of back sights: \(BS_L + BS_M = 1.5 + 1.8 = 3.300\) m. This matches (P).
Sum of fore sights: \(FS_M + FS_N = 2.0 + 1.0 = 3.000\) m. This matches (S).
RL of Point N: \(150.300\) m, found above. This matches (R).

Step 4: Why the other options are wrong.
Option (B) pairs (II) with (Q) = 2.500 m, but the actual sum of fore sights works out to 3.000 m, not 2.500 m, so this option is wrong. Options (C) and (D) pair (I) with (S) = 3.000 m, but the sum of BACK sights is 3.300 m (3.000 m is the sum of FORE sights), so these options have Column 1 and Column 2 mixed up for that row, and they also give RL of N as 150.500 m (T), which does not match the calculated 150.300 m.

Final Answer:
Sum of Back Sights = 3.300 m (P), Sum of Fore Sights = 3.000 m (S), RL of N = 150.300 m (R).
\[ \boxed{(I)-(P)\ ;\ (II)-(S)\ ;\ (III)-(R)} \]